I saw this proof:
Let S be the sample space. $S = \{s_1,s_2,...\} = \bigcup\limits_{i=1}^{\infty} \{s_i\}$
Suppose all outcomes are equally likely, $\Rightarrow$ $P(\{s_i\})=c>0$
$P(S) = P(\bigcup\limits_{i=1}^{\infty}\{s_i\}) = \sum\limits_{i=1}^{\infty}P\{s_i\} = \sum\limits_{i=1}^{\infty}c = \infty$, which contradicts that P(S)= 1. Hence, not all outcomes are equally likely.
I would like to ask why we can apply the axiom that $ P(\bigcup\limits_{i=1}^{\infty}\{s_i\}) = \sum\limits_{i=1}^{\infty}P\{s_i\} $ here, since I thought this axiom only works for mutually exclusive events, and the question did not state that the events are mutually exclusive.
Edit: spelling error